Abstract
AbstractWe study the dimension of loci of special line bundles on stable curves and for a fixed semistable multidegree. In case of total degree $$d = g - 1$$
d
=
g
-
1
, we characterize when the effective locus gives a Theta divisor. In case of degree $$g - 2$$
g
-
2
and g, we show that the locus is either empty or has the expected dimension. This leads to a new characterization of semistability in these degrees. In the remaining cases, we show that the special locus has codimension at least 2. If the multidegree in addition is non-negative on each irreducible component of the curve, we show that the special locus contains an irrreducible component of expected dimension.
Funder
Israel Science Foundation
Gottfried Wilhelm Leibniz Universität Hannover
Publisher
Springer Science and Business Media LLC
Reference18 articles.
1. Alexeev, V., Birkenhake, C., Hulek, K.: Degenerations of Prym varieties. J. Reine Angew. Math. 553, 73–116 (2002)
2. Arbarello, E., Cornalba, M., Griffiths, P. A., Harris, J.: Geometry of algebraic curves. Volume I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267. Springer, New York (1985)
3. Alexeev, V.: Compactified Jacobians and Torelli map. Publ. Res. Inst. Math. Sci. 40(4), 1241–1265 (2004)
4. Beauville, A.: Prym varieties and the Schottky problem. Invent. Math. 41(2), 149–196 (1977)
5. Caporaso, L.: A compactification of the universal Picard variety over the moduli space of stable curves. J. Am. Math. Soc. 7(3), 589–660 (1994)